We obtain effective bounds on the heights of algebraic integers whose orbits contain multiplicatively dependent values modulo S-integers. Our method is based on a new upper bound on the so-called S-height of polynomial values over the ring of integers of $\mathbb{K}$. Our results provide an effective variant of a recent result of A.Bérczes, A.Ostafe, I.E.Shparlinski and J.H.Silverman (arXiv:1811.04971) on multiplicative dependence modulo a finitely generated subgroup by eliminating the use of non-effective results by K.F.Roth and G.Faltings.
We prove the density of the sets of the form ${λ_1^m μ_1^n ξ_1 +...+λ_k^m μ_k^n ξ_k : m,n \in \mathbb N}$ modulo one, where $λ_i$ and $μ_i$ are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelean groups.
This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies that the only consecutive pairs of multiplicatively dependent integers larger than 1 are $(2,8)$ and $(3,9)$. For triples, we prove the following theorem: If $a\notin \{2,8\}$ is a fixed integer larger than 1, then there are only finitely many triples $(a,b,c)$ of pairwise distinct integers larger than 1 such that $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ are each multiplicatively dependent. Moreover, these triple…
In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for various cases, although in general there is a logarithmic gap between lower and upper bounds.
Let $ζ_k$ be a $k$-th primitive root of unity, $m\geqϕ(k)+1$ an integer and $Φ_k(X)\in\mathbb Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+ζ_k,\mathcal N)$ is a canonical number system, with $\mathcal N=\{0,1,\dots,|Φ_k(m)|\}$. Moreover we also discuss whether the two bases $-m+ζ_k$ and $-n+ζ_k$ are multiplicatively independent for positive integers $m$ and $n$ and $k$ fixed.
Recently the authors showed that the algebraic integers of the form $-m+ζ_k$ are bases of a canonical number system of $\mathbb{Z}[ζ_k]$ provided $m\geq ϕ(k)+1$, where $ζ_k$ denotes a $k$-th primitive root of unity and $ϕ$ is Euler's totient function. In this paper we are interested in the questions whether two bases $-m+ζ_k$ and $-n+ζ_k$ are multiplicatively independent. We show the multiplicative independence in case that $0<|m-n|<10^6$ and $|m|,|n|> 1$.
In this paper, we first prove that given pairwise distinct algebraic numbers $α_1, \ldots, α_n$, the numbers $α_1+t, \ldots, α_n+t$ are multiplicatively independent for all sufficiently large integers $t$. Then, for a pair $(a,b)$ of distinct integers, we study how many pairs $(a+t,b+t)$ are multiplicatively dependent when $t$ runs through the integers. For such a pair $(a,b)$ with $b-a=30$ we show that there are $13$ integers $t$ for which the pair $(a+t,b+t)$ is multiplicatively dependent. We conjecture that $13$ is the largest value of such translations for any $(a,b)$, where $a \ne b$, pro…
Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.
Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, we prove that for arbitrary $k-$regular partiton function $b_k(n)$ and integer $M$ coprime to $6$, there are infinitely many Ramanujan-type congruences of $b_k(n)$ modulo $M$.
We give an infinite family of polynomials that have roots modulo every positive integer but fail to have rational roots. Each polynomial in this family is made up of monic quadratic factors that do not have linear term.
We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for almost all $n$. This study was inspired by similar congruences modulo 4 in the work by the second author and Garvan.
We study the uniform distribution of the polynomial sequence $λ(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $λ(P)$ is uniformly distributed in $\mathbb{Z}$ if and only if $P(x)$ has at least one irrational coefficient other than the constant term. In the case of even degree, we prove a stronger result: $λ(P)$ intersects every congruence class modulo every integer if and only if $P(x)$ has at least one irrational coefficient other than the constant term.
Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N^(1-ε) for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p^(1-ε) for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N^(1-ε) for all N in a density one subset of the integers.
Let $r$ and $s$ be multiplicatively independent positive integers. We establish that the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.
Denoting by V(n) the number of regular integers $(\text{mod}n)$, we study properties of the sequences $\displaystyle \biggl(\frac{V(n+1)}{V(n)}\biggr)_{n\geq1}$ and $\displaystyle (V(n+1)-V(n))_{n\geq1}$. We also prove that the sets $\displaystyle\biggl \lbrace\frac{ψ(n)}{V(n)} : n\geq1\biggr \rbrace$ and $\displaystyle\biggl \lbrace\frac{V(n)}{ϕ(n)} : n\geq1\biggr \rbrace$ are everywhere dense in $(1,+\infty)$.
For an element $a$ of an integral domain D under an equivalence relation τ, the τ-factorization of a is defined as λa_1 a_2... a_k, where λis a unit in D and a_i τa_j for all i, j. An irreducible element has no proper τ-factorization; that is, a τ-factorization in which there is more than one distinct non-unit factor. In this paper, the irreducible integers under the congruence modulo n relation for some values of n are found, and these findings are generalized in the first step toward a general characterization of the irreducible integers under this relation for any prime n.
For any non-negative integer $n$ and non-zero integer $r$, let $p_r(n)$ denote Ramanujan's general partition function. By employing $q$-identities, we prove some new Ramanujan-type congruences modulo 5 for $p_r(n)$ for $r=-(5λ+1), -(5λ+3), -(5λ+4), -(25λ+1)$, $-(25λ+2)$, and any integer $λ$.
We study variants of the Erd\H os distance problem and dot products problem in the setting of the integers modulo $q$, where $q = p^{\ell}$ is a power of an odd prime.
We study the algebraic dynamical systems generated by triangular systems of rational functions and estimate the height growth of iterations generated by such systems. Further, using a result on the reduction modulo primes of systems of multivariate polynomials over the integers, we study the periodic points and the intersection of orbits of such dynamical systems over finite fields. 1
Prempreesuk, Noppakaew, and Pongsriiam determined the Zeckendorf representation of the multiplicative inverse of $2$ modulo $F_n$, for every positive integer $n$ not divisible by $3$, where $F_n$ denotes the $n$th Fibonacci number. We determine the Zeckendorf representation of the multiplicative inverse of $a$ modulo $F_n$, for every fixed integer $a \geq 3$ and for all positive integers $n$ with $\gcd(a, F_n) = 1$. Our proof makes use of the so-called base-$φ$ expansion of real numbers.
A polynomial with coefficients in the ring of integers $\mathcal{O}_{K}$ of a global field $K$ is called intersective if it has a root modulo every finite-indexed subgroup of $\mathcal{O}_{K}$. We prove two criteria for a polynomial $f(x)\in\mathcal{O}_{K}[x]$ to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon $K$ and the polynomial $f$. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Ch…
Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer coefficients and a positive integer $n$, let $α(n)$ be the number of integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots, x_t)\equiv a\ (mod\ n)$ is solvable. We describe a method that allows to determine the function $α$ associated to polynomials of the form $c_1x_1^k+c_2x_2^k+\cdots+c_tx_t^k$. Then we apply this method to polynomials that involve sums and differences of squares, mainly to the polynomials $x^2+y^2, x^2-y^2$ and $x^2+y^2+z^2$.
Let $a$ and $m>0$ be integers. We show that for any integer $b$ relatively prime to $m$, the set $\{a^n+bn:\ n=1,\ldots,m^2\}$ contains a complete system of residues modulo $m$. We also pose several conjectures for further research; for example, we conjecture that any integer $n>1$ can be written as $k+m$ with $2^k+m$ prime, where $k$ and $m$ are positive integers.
Let $n=p_1^{ν_1}... p_r^{ν_r} >1$ be an integer. An integer $a$ is called regular (mod $n$) if there is an integer $x$ such that $a^2x\equiv a$ (mod $n$). Let $\varrho(n)$ denote the number of regular integers $a$ (mod $n$) such that $1\le a\le n$. Here $\varrho(n)=(ϕ(p_1^{ν_1})+1)... (ϕ(p_r^{ν_r})+1)$, where $ϕ(n)$ is the Euler function. In this paper we first summarize some basic properties of regular integers (mod $n$). Then in order to compare the rates of growth of the functions $\varrho(n)$ and $ϕ(n)$ we investigate the average orders and the extremal orders of the functions $\varrho(n)/…
Let F be a finite field and let b and N be integers. We prove explicit estimates for the probability that the number of rational points on a randomly chosen elliptic curve E over F equals b modulo N. The underlying tool is an equidistribution result on the action of Frobenius on the N-torsion subgroup of E. Our results subsume and extend previous work by Achter and Gekeler.
A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its product is congruent to $μ(p-1)$ modulo $p$, where $μ$ is the Möbius function. In this paper, we will generalize these two interesting congruences and give the congruences of the sum and the product of integers with the same index modulo $n$.
Periods modulo $p$ of integer sequences associated with division polynomials of genus $2$ curves
Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；Department of Mathematics, Faculty of Science, Kyoto University（京都大学理学部数学科）；Department of Mathematics, Cooperative Faculty of Education, Gunma University（群马大学教育学部协同数学科）；Department of Mathematics, Graduate School of Science, Kobe University（神户大学理学研究数理学科）；Department of Mathematical Sciences, Graduate School of Science, Tokyo Metropolitan University（东京都立大学理学研究科数学科学系）
We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic progression of length |2S|-|S|+1. This is the first result of this nature which places no unnecessary restrictions on the size of S.
Let α be a prime Hurwitz integer. Hα, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of H, which is the set of all Hurwitz integers. We consider left congruent module α and, the domain of related modulo function in this study is ZN(α), which is residual class ring of ordinary integers with N(α) elements, which is the norm of prime Hurwitz integer α. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. Thereby,…
An integer $k$ is called regular (mod $n$) if there exists an integer $x$ such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$ possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that $k^{m+1} \equiv k$ (mod $n$). Let $\varrho(n)$ denote the number of regular integers (mod $n$) in the set $\{1,2,\ldots,n\}$. This is an analogue of Euler's $ϕ$ function. We introduce the multidimensional generalization of $\varrho$, which is the analogue of Jordan's function. We establish identities for the power sums of regular integers (mod $n$) and for some other finite…
In this paper, we show that for almost all primes p there is an integer solution x in [2,p-1] to the congruence x^x == x mod p. The solutions can be interpretated as fixed points of the map x -> x^x mod p, and we study numerically and discuss some unexpected properties of the dynamical system associated with this map.
For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods for determining if there exists infinitely many primes $p$ such that the index of the reduction of $W_i$ modulo $p$ divides $k_i$ and such that the Artin symbol of $p$ on $F$ is contained in $C$. The results are a multivariable generalization of H.W. Lenstra's work. As an application, we determine all integers $a_1, \ldots , a_n$ such that $\text{ord}_p(a_1) =…
For any $\varepsilon >0$, we obtain an asymptotic formula for the number of solutions $n \le x$ to $$ \lVert αn + β\rVert < x^{-\frac{1}{4}+\varepsilon} $$ where $n$ is $[y,z]$-smooth for infinitely many real number $x$. In addition, we also establish an asymptotic formula with an additional square-free condition on $n$. Moreover, if $α$ is quadratic irrational then the asymptotic formulas holds for all sufficiently large $x$. Our ingredients come from the Harman sieve which we adapt suitably to sieve for $[y,z]$-smooth numbers. The arithmetic information comes from estimates for exponential s…
Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we obtain new results concerning the number of solutions to congruences of the form $$y\equiv f(x) \pmod{q},$$ with integer variables lying in some cube $\mathcal B$ of side length $H$. Our argument uses ideas of Cilleruelo, Garaev, Ostafe and Shparlinski which reduces the problem to the Vinogradov mean value theorem and a lattice point counting problem. We treat the lattice point problem differently using transference principles from the Geometr…
A sharp bound is obtained for the number of ways to express the monomial $X^n$ as a product of linear factors over $\mathbb{Z}/p^α\mathbb{Z}$. The proof relies on an induction-on-scale procedure which is used to estimate the number of solutions to a certain system of polynomial congruences. The method also applies to more general systems of polynomial congruences that satisfy a non-degeneracy hypothesis.
For a nonnegative integer $n$, and a prime $\wp$ in $\mathbb{F}_q[T]$, we prove a result that provides a method for computing the number of integers $m$ with $0 \le m \le n$ for which the Carlitz binomial coefficients $\binom{n}{m}_C$ fall into each of the residue classes modulo $\wp$. Our main result can be viewed as a function field analogue of the Garfield-Wilf theorem.
We study the qKZ difference equations with values in the $n$-th tensor power of the vector $sl_2$ representation $V$, variables $z_1,\dots,z_n$ and integer step $κ$. For any integer $N$ relatively prime to the step $κ$, we construct a family of polynomials $f_r(z)$ in variables $z_1,\dots,z_n$ with values in $V^{\otimes n}$ such that the coordinates of these polynomials with respect to the standard basis of $V^{\otimes n}$ are polynomials with integer coefficients. We show that the polynomials $f_r(z)$ satisfy the qKZ equations modulo $N$. Polynomials $f_r(z)$ are modulo $N$ analogs of the hyp…
We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a solution for every $x \in \mathbb{Z}_n$? Using only elementary methods, we answer fully this question for exponents $k \leq 10$, and we further discuss some intermediary cases such as categorizing the values of $n$ such that every element in $\mathbb{Z}_n$ can be written as a sum of three squares. Hensel's Theorem for $p$-adic integers plays a key role. Final…
Let $\mathbb{Z}^{ab}$ be the ring of integers of $\mathbb{Q}^{ab}$, the maximal abelian extension of $\mathbb{Q}$. We show that there exists an algorithm to decide whether a system of equations and inequations, with integer coefficients, has a solution in $\mathbb{Z}^{ab}$ modulo every rational prime.
Given a finite set of bases $b_1$, $b_2$, \dots, $b_r$ (integers greater than $1$), a multi-base representation of an integer~$n$ is a sum with summands $db_1^{α_1}b_2^{α_2} \cdots b_r^{α_r}$, where the $α_j$ are nonnegative integers and the digits $d$ are taken from a fixed finite set. We consider multi-base representations with at least two bases that are multiplicatively independent. Our main result states that the order of magnitude of the minimal Hamming weight of an integer~$n$, i.e., the minimal number of nonzero summands in a representation of~$n$, is $\log n / (\log \log n)$. This is …
